A note on non-empty cross-intersecting families

The families $F_1\subseteq \left(\frac{\left[n\right]}{k_1}\right),F_2\subseteq \left(\frac{\left[n\right]}{k_2}\right),\dots ,F_r\subseteq \left(\frac{\left[n\right]}{k_r}\right)$ are said to be cross-intersecting if $|F_i\cap F_j|⩾1$ for any $1⩽i<j⩽r$ and $F_i\in F_i$, $F_j\in F_j$. Cross-intersecting families $F_1,F_2,\dots ,F_r$ are said to be non-empty if $F_i\ne 0̸$ for any $1⩽i⩽r$. This paper shows that if $F_1\subseteq \left(\frac{\left[n\right]}{k_1}\right),F_2\subseteq \left(\frac{\left[n\right]}{k_2}\right),\dots ,F_r\subseteq \left(\frac{\left[n\right]}{k_r}\right)$ are non-empty cross-intersecting families with $k_1⩾k_2⩾\cdots ⩾k_r$ and $n⩾k_1+k_2$, then ${\sum }_{i=1}^r\left|F_i\right|⩽max\{\left(\frac{n}{k_1}\right)-\left(\frac{n-k_r}{k_1}\right)+{\sum }_{i=2}^r\left(\frac{n-k_r}{k_i-k_r}\right),{\sum }_{i=1}^r\left(\frac{n-1}{k_i-1}\right)\}$. This solves a problem posed by Shi, Frankl and Qian recently. The extremal families attaining the upper bounds are also characterized.